Thread: Trying to work through a Fibonacci problem using Binet's Formula

I am trying to show that the square of any Fibonacci Number differs by one from the product of its neighbors:





My best attempt is to substitute Binet's Formula:





and simplify, but keep getting buggered near the end. I get down to:





and can go no further due to the unequal bases. I know (because even values of "n" yield -1) that:





which would make the final formula:





but I seem to be missing something. I could swear I've seen this done, but can't find it anywhere. Any pointers as to what I'm missing?


- Stephen

I realized this morning that I had overlooked some of my algebra:

and

So, moving through a few more steps I get to:



I'm not sure if it's any better, but it's something ...

Wikipedia has a nice proof of this theorem (Cassini's identity):

Cassini and Catalan identities - Wikipedia, the free encyclopedia

That is nice - I was hoping for something algebraic, but this is about as straightforward as I could ask for. I have two concerns with this solution though:

1. Using a matrix looks an awful lot like we simply restated the problem and called it solved
2. While creating the matrix using with an odd value of n works well, won't using an even value of n yield a determinant of which will always be positive?

It looks like I'll have to brush up on matrices so I can explain it. Wikipedia to the rescue once again!

Thank you!

- Stephen

1. But you are definitely allowed to write as the determinant of a 2x2 matrix, right?

2. Yes, but for even n.

Yes on both counts. My "concerns" were rather poorly worded - what I meant to convey is that I'm worried my unfamiliarity with matrices leaves me unprepared to answer what should be a couple of simple questions, not that there was a problem with the proof. Hence my need to "brush up."

- Stephen